Problem — Speculative physics: double gravity

Speculative physics: double gravity. If $g$ doubled, how would the potential energy, the fall speed and the maximum height of a jump change? Discuss.

Solution

Analyse the three quantities as functions of $g$, keeping the mass $m$, the height $h$ and the initial jump speed $v_0$ fixed.

Potential energy. $E_p = mgh$ is linear in $g$: if $g \to 2g$,

$E_p' = m(2g)h = 2\,mgh = 2\,E_p \quad\Rightarrow\quad \text{doubles}$

Fall speed. From $mgh = \tfrac12 mv^2$ it follows that $v_\text{fall} = \sqrt{2gh}$, which goes as $\sqrt{g}$:

$v' = \sqrt{2(2g)h} = \sqrt{2}\cdot\sqrt{2gh} = \sqrt{2}\;v_\text{fall} \quad\Rightarrow\quad \times\sqrt{2} \approx \times 1{,}41$

Maximum jump height. Given the initial push $v_0$, the body rises until $\tfrac12 mv_0^2 = mg\,h_\text{max}$, i.e. $h_\text{max} = v_0^2/(2g)$, which goes as $1/g$:

$h_\text{max}' = \frac{v_0^2}{2(2g)} = \frac{1}{2}\cdot\frac{v_0^2}{2g} = \tfrac12\,h_\text{max} \quad\Rightarrow\quad \text{is halved}$

$\ev{E_p:\ \times 2; \qquad v_\text{fall}:\ \times\sqrt{2}; \qquad h_\text{max}:\ \times\tfrac12}$

Physical reading. A world with double gravity would be much more hostile to movement: you would jump half as high and falls would be much more violent (impact energy $E_p$ doubled, speed $\times 1{,}41$). The biology of animal movement — bones, muscles, gaits — would be thoroughly upended.

Collegamenti

Argomenti: Work and energy Concetti: Gravitational potential energy Competenze: Analysis of limiting cases and speculative physics